A hairspring is a key component in a mechanical timepiece. A hairspring is one of the two main components of an oscillator of a timepiece, the other being the balance wheel. The oscillator provides the means of time regulation via its simple harmonic motion.
A balance wheel acts as the inertial element, and is engaged with the inner terminal of a spiral-shaped hairspring. The spiral geometry of a hairspring is generally provided in the form of an Archimedean spiral, generally having a constant pitch. The outer terminal of the hairspring is generally fixedly attached to a fixed stud.
Ideally, the hairspring provides a restoring torque to the balance wheel that is proportional to the wheel's displacement from an equilibrium position, and equations of motion may be utilised to describe a linear second-order system thereof. The equilibrium position of an oscillator is defined as the angular position of the balance wheel such that when the balance wheel is static, that is when the net torque applied by the hairspring to the balance wheel is zero. The resulting oscillator is isochronous, this meaning its natural frequency is independent of its amplitude.
Being isochronous is an important property for an oscillator used in a timepiece as it requires regular torque input from an escapement to compensate for dissipative effects of friction. The torque provided by the escapement may not be constant due to a number of factors, which directly affects the oscillator amplitude. As such, an isochronous oscillator provides a more reliable and stable time regulation.
Typically, the spiral turnings of a hairspring for a timepiece are maintained as concentric as possible when the balance wheel rotates about its equilibrium position for reasons including:    (i) a hairspring that is not concentric does not have its centre of mass located close to the axis of rotation. As the balance wheel rotates, the center of mass may wander in such a way as to generate a radial force that is compensated by bearings, resulting in excessive friction;    (ii) A hairspring that is not concentric also has a geometry that deviates from an Archimedean spiral during operation, which results in a nonlinear second-order system that is not isochronous; and    (iii) In some cases, a hairspring that is not concentric may significantly distort its spiral geometry such that the adjacent turnings collide and damage each other, as well resulting is a system that is not isochronous.
Within the prior art, hairspring concentricity may be improved by modifying the geometry of the inner and outer terminal curves based on Phillips and Lossier mathematical models for hairspring design.
Breguet has implementing such theories in its Breguet over-coil for the outer terminal. The over-coil uses a modified outermost turning which is raised and curved inwardly. However, this method can only maintain partial concentricity, and production the required shape in the outermost turning increases manufacturing difficulties and costs.
Another method of the prior art to increase hairspring concentricity is to selectively stiffen sections of the hairspring strip first proposed by Emile and Gaston Michel in the 1958 article entitled “Spiraux plats concentriques sans courbes”ly (Concentric flat hairsprings without curves), published de by Societe Suisse Chronometrie.
The authors discovered via trial and error that hairspring concentricity may be improved by stiffening a section of the hairspring using an angle strip. Difficulties with such a hairspring include difficulty in mass production, and such a hairspring remains an academic curiosity.
Also within the prior art, Patek Philippe stiffened a hairspring section in its Spiromax hairspring using a strip of variable width to achieve the stiffening effect. Patek Philippe also developed and patented a design methodology (patent number EP 03009603.6) by calculating the location of the center of mass when the hairspring is relaxed. The stiffening is achieved design by a widening of the outer side on the outermost turning of the hairspring.
To maintain a hairspring as isochronous, hairspring design requires insensitivity to temperature variations. The Young's modulus of a material which its stiffness typically varies slightly with temperature.
In a hairspring, the Young's modulus determines the spring constant and ultimately the natural frequency of the oscillator. Any variation of the hairspring's Young's modulus with temperature will negatively impact the oscillator's ability to reliably regulate time.
A problem of the Young's modulus's sensitivity to temperature in modern hairsprings has been widely addressed by the use of Nivarox in the manufacture of hairsprings. Nivarox is a metallic alloy having a Young's modulus that is extremely low, but not zero, in respect of sensitivity to temperature variations.
The advent of micro-fabrication and the use of silicon in the watch industry has over the past decade introduced new methods to design and manufacture of hairsprings with improved isochronism. Such technology allows the manufacture of hairspring based on variations of the strip width to selectively modify the spring's bending stiffness along its entire arc length.
Further, such technology allows the prospect of achieving a hairspring whose Young's modulus is completely insensitive to temperature variations. The process of de-sensitizing the hairspring's Young's modulus with respect to temperature variation is defined as thermo-compensation.
Manufacture of a hairspring having a variable strip width is only practically possible utilising micro-fabrication technology due to its ability to manufacture any planar component to high precision.
Hairspring concentricity may be increased utilising micro-fabrication techniques based on theory, numerical simulation, or experimentation. The Patek Philippe Spiromax is an example of a silicon hairspring with a section of increased strip width in the outermost turning near the outer terminal, placed and sized to increase hairspring concentricity.
Micro-fabrication technology may also allow application of a thin coat of silicon dioxide on a silicon hairspring for thermo-compensation purposes. The Young's modulus of silicon decreases with rise in temperature while that of silicon dioxide tends to increase.
Therefore, by the precise application of silicon dioxide coating of the correct thickness onto a silicon bulk, it is possible to produce a composite hairspring where the thermal sensitivities of the Young's modulus of the two materials substantially cancel each other. This may result in a hairspring with an overall Young's modulus that is theoretically insensitive to temperature variations.